Bring a large pot of water to boil for the egg noodles.
Cream of Mushroom Sauce: heat a medium sauce pot over medium heat. Add extra virgin olive oil and butter. When butter melts, add in chopped button mushrooms and cook 5 minutes, until just tender. Sprinkle in flour and cook 1 minute. Whisk in chicken stock and bring to a bubble then stir in the whole milk or cream. Reduce heat to low and gently simmer. Season the sauce with nutmeg, salt and pepper.
Heat a nonstick skillet over medim high heat. Add 2 tablespoons extra virgin olive oil, then the shallots and mushrooms. Cook the mushrooms 8 minutes, until tender. Season with thyme, salt and pepper and deglaze the pan with 1/3 cup dry white wine. Reduce heat to medium low and let the liquid cook off.
Preheat the broiler to high.
While the mushrooms cook, drop egg noodles in water. Salt the water and cook noodles to al dente. Drain noodles and return to hot pot. Add the sauce to the pot and toss noodles to coat in sauce.
Lightly cover a casserole dish with softened butter then transfer cream of mushroom noodles to the casserole dish and top with the wild mushroom mixture and the shredded cheese. Place the casserole under the broiler and heat until cheese is brown at the edges and bubbly. Garnish with chives, if desired.